 mersenneforum.org > Math What is the probability distribution for M42 ?
 Register FAQ Search Today's Posts Mark Forums Read 2004-05-31, 18:08 #1 dsouza123   Sep 2002 29616 Posts What is the probability distribution for M42 ? The following page, Where is the next Mersenne prime? http://www.utm.edu/research/primes/n...tMersenne.html has a graph, Probability distribution for t = log2(log2(41st Mersenne)) http://www.utm.edu/research/primes/gifs/M41_none.gif just after the text: If we had no idea what the first 38 Mersennes were, we would predict values for t=log2(log2(Mn)) as follows. M41 was found very near the highest probability. What is the graph for M42 ? How is it calculated ?   2004-06-01, 16:58   #2
R.D. Silverman

Nov 2003

22·5·373 Posts Quote:
 Originally Posted by dsouza123 The following page, Where is the next Mersenne prime? http://www.utm.edu/research/primes/n...tMersenne.html has a graph, Probability distribution for t = log2(log2(41st Mersenne)) http://www.utm.edu/research/primes/gifs/M41_none.gif just after the text: If we had no idea what the first 38 Mersennes were, we would predict values for t=log2(log2(Mn)) as follows. M41 was found very near the highest probability. What is the graph for M42 ? How is it calculated ?
No offense intended, but your entire line of questioning is "wrong headed".
I'm not sure that you really know what it is that you are asking for.

"near the highest probability" is mathematical gibberish. It is *believed*
that the Mersenne primes are Poisson distributed with respect to their
exponents. IN THE LONG RUN. But asking for a "graph for M42" (i.e. a specific number) is meaningless, especially in the SHORT RUN.

If instead you ask: "given that we know M41, p = 24036583 and that the
probability that a randomly chosen q, for q > p has M_q a prime, what value
of q is most likely to make M_q prime", the answer clearly is "the smallest
untested value of q greater than p".

If instead you ask: "given that the primes q that make M_q prime" are
Poisson distributed, and given that M_24036583 is prime, what is the
expected value for the next prime M_q", that is an entirely different
question.

There are heuristics which suggest that the expected number of Mersenne
primes in the interval [2^n, 2^2n] is exp(gamma), independent of n as
n-->oo. But there is no proof.

Thus, we expect about 1.7 Mersenne primes in the exponent interval [24M, 48M], but trying to guess which one is the "most likely" is meaningless.

All this guessing as to the next "most likely place to search" is meaningless.      2004-06-02, 02:16   #3
markr

"Mark"
Feb 2003
Sydney

3×191 Posts Quote:
 Originally Posted by dsouza123 The following page, Where is the next Mersenne prime? http://www.utm.edu/research/primes/n...tMersenne.html has a graph, Probability distribution for t = log2(log2(41st Mersenne)) http://www.utm.edu/research/primes/gifs/M41_none.gif just after the text: If we had no idea what the first 38 Mersennes were, we would predict values for t=log2(log2(Mn)) as follows.
The catch is the phrase, "If we had no idea what the first 38 [41?] Mersenne primes were". IF we were in this sad position, AND for some reason we wanted to find the 42nd, a graph like that shows where to look.

However, we know more than that - we know the first 41, give or take a bit of uncertainty. After the graph referred to, Chris Calwell's page goes on to explain (with another graph) that the best place to look is just after the largest one we already know. Basically it is because the distribution is modelled so well by a Poisson process, which has "no memory" of what came before.

I hope that helps!

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